Embed Size (px)
Citation preview
HAL Id: hal-00945782https://hal.archives-ouvertes.fr/hal-00945782
Preprint submitted on 12 Feb 2014
HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.
In defense of a non-newtonian economic analysisDiana Filip, Cyrille Piatecki
To cite this version:
Diana Filip, Cyrille Piatecki. In defense of a non-newtonian economic analysis. 2014. �hal-00945782�
In defense of a non-newtonianeconomic analysis∗
Diana Andrada Filip† & Cyrille Piatecki‡
October 8, 2012
Abstract
The double-entry bookkeeping promoted by Luca Pacioli in the fif-teenth century could be considered a strong argument in behalf of themultiplicative calculus which can be developed from the Grossmanand Katz non-newtonian calculus concept. In order to emphasize thisstatement we present a brief history of the accountancy in its earlytime and we make the point of Ellerman’s research concerning thedouble-entry bookkeeping.
Contents
1 Introduction
According to Kuhn (1962), a paradigm shift is a radical change of our wayto understand the world such that no one inside the field of reflectionwhere it occurs can ever refers to the corpus that was pregnant before theshift. But, for a paradigm shift to occur, the failure of the older paradigm
∗This work was supported by CNCSIS - UEFISCSU, project number PNII IDEI2366/2008†Babes-Bolyai University of Cluj-Napoca, Romania and LEO, Orleans University,
France — [email protected].‡LEO, Orleans University, France — [email protected]
1
must be acknowledged by a scientific community. And to acknowledgethat there are failures, a scientific community must not be locked-in a wayto look at some phenomenons or in a way to analyse those phenomenons.
And in complete opposition to what some could think, it is through ourmodel of the world that we try to understand it because nobody has neverbeen able to collect data without a conception of what are the data.
Now, because our cerebral enginery has evolved in such a way that, atbirth, we are only able to conceive elementary additions and substractions,all other operations being the object of a complex process of learning —see for example Lakoff & Nunez (2000) —, we are locked in an additiveconception of our world which is rarely questioned.
2 The accounting locked-in
It is hard to imagined that the way we take record of our transactions couldhave been managed in a different way, better adapted to the description ofthe growth or decline of ours operations.
2.1 A brief accountancy history in its early time
It is acknowledge that human beings count since its emergence on thisworld. But a serious accountancy system like the double entry book-keeping appears only in the 14th century Italy and not in the merchantcivilisation of the middle east or in Greece or in Rome. According to Little-ton (1933), there are 7 key ingredients which led to the creation of a doubleentry book keeping :
① Existence of a private property system : This is a mandatory ingredientbecause bookkeeping is concerned with the record properties, oftransfer of properties and on property rights.
② Accumulation of capital : the growth of the human activity has beeneffective only when those with a know-how has been able to borrowresources to those who possessed them in such a way that commerceand credit ceased to be trivial.
③ Commerce at a widespread level : At a local level, small volume tradingdoes not create a pressure to organise a strong system of bookkeep-
2
ing, because a simple accountancy is perfectly sufficient for smallquantities transactions.
④ Existence inter-personal credit with an enforcement guarantee : If all trans-actions are untied on the spot, there is no incentive to keep any record.The enforcement guarantee need as a precondition the existence ofan authority strong enough to sanction the non-payment of the prin-cipal and the interest due to the loaners. As is shown by the earlycivil code as the Ur-Numu or the Hamuraby ones — nearly 2285B.C. —, the pre-existence of the State, if not a mandatory condition isnecessary to the realisation of this ingredient.
⑤ Writting : Obviously, if one cannot write, because human memoryis too fallible, there is no way to record any thing. The oral transferof information from people to people cannot offer any guaranties ofauthenticity.
⑥ Money : It is also a mandatory condition, because without a commondenominator, bookkeeping is nearly impossible. In the contrary, withmoney and from the point of view of bookkeeping, transactions areno more that a set of monetary values.
⑦ An arithmetic : This is also mandatory because without the masteryof an arithmetic there is no way to compute the monetary details ofthe transactions.
As signaled by Alexander (2002), many of these factors where presentlong before the 14th century but either there where not present in the sametime and place or they where present in an unstructured form or with a notstrong enough pregnancy. Therefore, in what concern writing, if it is not anecessary condition for civilization — the gallic civilisation is acknowledgeas a civilization even if it never acquired writing —, it was mandatory tobegin historical times since History could not exist without records. Buteven if the beginnings of arithmetic are contemporary of the civilization,arithmetic understood as the systematic manipulation of numbers, wasnot a tool acquired on a sufficient scale until the middle-age to help to thedevelopment of a double entry bookkeeping.
More than that, the nearly universal use of roman numeral in the occi-dental world long after that the arabic numeral have been introduced, has
3
been a severe restraining factor because of the non existence of the zero inthe roman numeral systems. Yet, a neutral element, and zero is the neutralelement of the addition, is mandatory to the development of a double entrybookkeeping as would be shown in section 2.2.
Nevertheless, in the ancient times, the problems encountered by mer-cantile or even States where alike with ours. For instance, because of taxcollecting, governments had strong incentives to keep records of receiptsand expenditures. As rich peoples often hired agents or used slaves toperform their operations, they needed to realize audits to verify the hon-esty and/or skill of their factotum. But the illiteracy and the cost of writing— ink and parchment — was so high, and the monetary systems so in-consistent, that only in the case where transactions were incredibly large,could we imagine to record them.
Nevertheless, because of he first prosperity times in the mankind his-tory in the area between the Tiger and the Euphrates rivers there was aneed to record transaction. This need was early codified. For instancein the Hammurabi code, it was required that an agent who was sellinga good in name of his principal gave him a price quotation under seal,the failure to perform this obligation being an invalidation argument forthe transaction. So, transactions were engraved in clay tablets which weresafely kept until final outcome and the recycled for a new transaction.
Egyptians use for a while the same support until the introduction of thepapyrus which permits easily to extend records. A profession of special-ized scribes was organized. It developed an elaborate internal verificationsystem whose honesty and credibility were enforced by royal audits whichconduct to important penalty in case of irregularities.
Ancient greeks introduced two major innovations : first, they estab-lished public accountants 1 to impose the authority and control of theState; secondly, about 600 B.C., they introduced coined money which notonly facilitated transactions but lighten bookkeeping operations by theintroduction of a local common unit. Under those innovations, the bank-ing system, which has existed since the old sumerian times, reached alevel never reached, allowing change and loan operations and even cashtransfers on a scale never reach until this time.
The romans developed the first system which recurrently maintain, atthe level of the households their daily receipts and the expenses in an ad-
110 chosen by lot.
4
versaria or account book. Then, they aggregated monthly those statementsin a cashbook known as a codex accepti et expensi. All those operations,developed because the assets and liabilities of the citizens where the basisof the taxation systems and eventually were used to determine civil rights.
Then roman accountants developed an elaborate system of checks andbalances for governmental receipts and spending which was necessary tofixed and verify all the operations links to a conquerors nation. Later on,when the empire was well established an accounting system was manda-tory to keep record of all level of fiscal operations. To coordinate all thepublic financial operation, they conceive the first annual budgets.
But at the time of the fall of the roman empire in 582, the necessarystructures needed to teach how to write an account has vanished. Onemust know that romans use nearly the same educational system which wasnearly universally used in Greece. Until 12, young upper class boys stayedin their families where they received an education where the emphasiswas set on letters, music and a great proportion of elementary arithmetic— essentially taught to know how counting either with the help of theirfingers or with the abacus2. Then, they were normally send in a schoolwhere they learned literature, grammar, some elements of logic, rhetoricand dialectics.
Only those who needed a deeper mathematical teaching, as the one whoplaned to become agrimensor, that is surveyors, used to learn geometry.If they planed to become architect they obeyed the Vitruve’s advices tolearn geometry, optics, arithmetic, astronomy and others fields as law,medicine, music, philosophy and history. Galen gave some nearby advicesfor student in medicine. . . The reason why mathematics where not highranked in the roman education was essentially that it was useful essentiallyfor liberal professions when the royal road in education conduced to publicfunctions.
But it seems that according to the accounting standards, quickly de-scribed above, this mathematical education was in all ways sufficient.
During the dark ages, because to learn you need peace, there was a fall ineduction in Europe to the unique exception of the british islands which, be-cause of their insolation, were protected against the quasi-universal chaos.
2It is not clear if the abacus is an roman invention or if it is a chinese one which comein occident as the silk cloth through the trade of the red sea and the coasting navigationalong india.
5
Of course, the higher centers of learning where rare but we do have testi-monies of the transmission of the roman educational system. Toward the9th century, in nearly every monasteries schools were organized to permitthe acquisition of the christian culture to those destined to the priesthood orto the cloistral life. But, because of the intended object of this eduction, thestudies were limited to reading, writing and the study of the Bible. Onlyin very rare places like the cathedral of York, mathematics were taught3.
Horrified by the low standard of education in continental Europe,Charlemagne, asked Alcuin to leave York to organize a court school withthe posted object to permit at least to the clerk to interpret the Holy Biblecorrectly.
But at his death, with the return of war, the educational level dropagainst until Gerbert was elected Pope under the name of Sylvestre II in 999.Gerbert was himself a mathematician who discovered some interestingdocument, so he favored the study of Boethius, one of the rare roman whostudied mathematics in the 5th century4. But, as Boethius was himselfinterested by the work of the surveyor, the mathematics studied had a morepronounced flavor of geometry than the needed arithmetics necessary tokeep the accounts.
At the end of the dark ages, the english church, fighting against thepagan influence, decided that mathematics had a too much paganist fla-vor and mathematics education again vanished. Even Sylvestre II mortalremains undergo the costs. Because of his efforts in the transmission of
3In 732, Egbert was bishop and head teacher of the York school. He organized thestudies in such a way to teach rhetoric, law — essentially canonic —, physics, arithmetic,geometry. As the date of Easter was changing according to the moon, some arithmeticswhere also taught.
4Until the beginning of the XXth century, Boethius has been credited to the first usageof the hindu-arabic numerals in Occident. But Smith & Karpinski (1911) have erasedthis credit, showing, by an erudite demonstration, that a man of his inquiring mindcouldn’t have let aside the least quantum of information that spreads out in the mediter-ranean world through commercial relations, but, as in his time, the zero has not yet beenincorporated as a place holder with the other numbers, the arabic ciphers where not animprovement in the calculation effort in comparison with roman numeral. So there whereno reason for Boethius to use them. In what concerns his Geometry, one of the three workswhich are attributed to Boethius, where one can found the complete hindu-arabic nu-merals including the zero, Smith & Karpinski (1911) have demonstrated that it has beenbelatedly falsified since, if Boethius have known such a system he would have used itmainly in his Arithmetics, that none of his disciples and successor never used it, and thatthe falsification of text was under complaint even in his own time.
6
the arabic science he obtains through his connection with the jews dur-ing his three years apprenticeship travel trough Catalogna and Cordoba,and, mainly, his introduction of the zero, a veil of sorcery stay associatedwith his name for sorcery until his exhumation in 1648 to verify if he wasstill inhabited by all the devils of inferno. Not the least reason of the sadpost-mortem history of Gerbert of Aurillac — Sylvestre II —, is his rolein revealing to the occidental intellectuals the, in that time relative new,hindu-arabic positional system of numerical notation and the fondamen-tal innovation attach to it that is to say the zero — initialy zephyrum —who was no more than the arabic shifr which itself was no more than theindian sunya which stand for void5. According to Turner (1951), the romannumerals which are, by nature, additives6 were even more easier to add orto subtract than the hindu-arabic ones. For instance, subtracting or adding126 — CXXVI — to 378 — CCCLXXVIII — is a very simple calculationwhich is perform without knowing any table as shown here after7 :
CCC LXX VIII (378)- C XX VI (126)
CC L V (255)
C XCCC LXX VIII (378)
+ C XX VI (126)D (C) (X)VI (504)
Al least, since Cajori (1950), the idea that multiplication and divisionwere truly hard problems if carry out with roman numeral has diffusedamong scholars. On that subject, reinforced by the fact that no multiplica-tive roman calculus has never been discovered in the document inheritedfrom roman or middle-age time, the common point of view was that thoseoperations were performed through the abacus. But, as shown by An-derson (1956), it’s a profound mistake8. For instance, in the case of themultiplication of 17 — XVII — by 60 — CL —, one has :
5On the transmission to Gerbert of the hindu-arabic numeral, one can consult Zuccato(2005).
6That is to say : 1977 or MCMLXXVII is, literally written, in roman numerals notationas 1 × 1000 + 9 × 100 + 7 × 10 + 1 × 7.
7X = VV, L = XXXXX and D = CCCCC.8In his appendix, Anderson (1956) take a brag pleasure to carry out the multiplication
of DCCXXIII by CCCLXIV reputed by Cajory as impossible to compute with romannumerals.
7
XVII× LX
D CCL L L+ C L X X
D CCCL C L X X
= D CCCCC X X
= D D X X
= M X X
176000
10201020
One can observe that the main difference between the two operating modeis that with roman numeral the operations are carried out from left to rightcontrary to the ones carried out with hindu-arabic numerals.
In what concern the division, there are no essential differences betweenthe two type of numeral. For instance, to compute 209 ÷ 11 that is —CCIX÷XI —, one has
C C I X X IC X X
LXXXX I X IXLXXXX I X
20 9 1111 199 9
0
But, as is not instantly apparent, division is in fact simpler with romannumerals than with hindu-arabic ones because it entitles the possibility tonot find the exact number of time the divisor will go in the dividend ateach step9. Anderson (1956) also shows how to calculate powers or how toextract square roots. This experimental archaeological explanation showsthat, contrary to the general conviction, it could be conjectured that it isnot because of its intrinsic complexity that roman numerals were finallyderelicted.
In the same time, one can conjecture that there was a disenchantmentbetween middle-age thinkers and roman numerals. If for scholars as Mur-ray (1991), the original sin of roman numerals was to embalm the primitive
9One must note that, in all those cases, there is a flavor a heriticism in the notationused conjointly with roman numeral since the sign + appears for the first time in a bookby Hodder in 1672, the sign × is a joint product of the algorithm by Napier in 1618 and ÷is due to Rahn in 1659. On those subjects, one can consult Smith (1958) and Cajori (1909).
8
principles of addition and substraction in such a way to block the entry,into notation, of the multiplication and the division, a point of view whichis dismissed by the former paragraphs, King (2001) sees a conjonction ofmany practical different causes. For example, the roman numeral were notfitted to accommodate great numbers because the letters used to constructa number were limited. As they used X for ten, C for one hundred and Mfor a thousand, they were also obliged to write a horizontal line throughor above the numerals to raised the value of the number by a factor of 10or of 1000. King (2001) displays a tentative by a certain Adriaen vanderGucht, realized in 1569, to construct a table of power of 10 from 1 to 29.But as one can observe, it couldn’t be used because of the ambiguity hereattached.
Number Representation
I 100 = 1 106 1012 1018 1024
X 101 107 1013 1019 1025
C 102 108 1014 1020 1026
M 103 109 1015 1021 1027
XM 104 1010 1016 1022 1028
CM 105 1011 1017 1023 1029
As shown by the anecdote retrieved from Suetonius’ Life of Twelve Cae-sars by Kaplan (2000), the roman numerals limited number of signs was atrue deficiency which could conduct to many legal resorts. When dying,Livie, August’s wife and certainly one of the most powerful women of theancient times, decided to bequest to Galba 50000000 sesterces, that is tosay, in roman numeral notations |D|. But, the son of Livie, the emperorTiberius, who doesn’t like Galba, insisted that |D| be read as 500000 becausequia notata non perscripta erat summa10.
This may seems a venal sin of roman numerals but, as shown by Murray(1991), as time goes by, with the help of the need for large numbers, romannumerals became truly impractical. For instance, Murray reports a 1649
10Because the sum was in notation, not written in full.
9
selling of the english crown lands for a amount of 1 423 710 pounds, 18shillings and 6 pence which have been noted by the exchequer
C M C h s dM iiij xxiij vij x xviij iij
Even for smaller numbers, during the 15th century the roman numeralsbecame inappropriate as a numbering system as one can realize in lookingto the 24 hour clock-face situated on the inside of the western wall of theduomo Santa Maria del Fiore in Florence.
A second problem was that scribes had introduced some variationswhich were very difficult to decipher as in some french manuscripts writtenbetween the 13th and the 15th century as :
XX•
IIII= 81, IXXXIII = 183, VIIXXVI = 156
as is shown in Lehmann (1936). An other point which renders caducthe usage of the roman numeral was that, since the introduction of thehindu-arabic numerals, peoples used to mixed them with the use of romannumeral. For instance Bergner (1905) — cited by King (2001) — brings outsome german inscriptions such as :
mcccc8 1408 stamp of an Augsburg religious dignitary1• 4 • Lxiii 1463 on a gravestone from Salzburg14XCIIII 1494 on the altar of St. Othmar in Naumburg1 • VC • V 1505 on a bell in Keila near Ziegenruck1 • VC • 6 1506 on a bell in Neustadt a. O.15X5 1515 in Lauffen near RottweillMD.25 1525 in the Schlosskirche in Chemnitz
And this degenerated usage was not the only appanage of germans,since even the great french humanist Guillaume Bude use to note the year1534 by M.5.34.
One couldn’t say if this out of favoring has been the origin of theintroduction of some to day forgotten numeral, but as shown by King(2001) that during the 10th century, John of Basingstoke, who was one of
10
the first in england to master greek, is reputed to have introduced somenew ciphers deemed of greek origin11. Those ciphers were very simplebut they were not able to note numbers greater than 99 as is shown in thefollowing table.
1 2 3 4 5 6 7 8 9
10 20 30 40 50 60 70 80 90
With such ciphers one can write very easily numbers as
55 62 99
Even if of limited range, this type of ciphers pleased at least the cis-tercian monks who designed some clever extensions. As shown by King(2001), cyphers of that type flourished in Europa. During the 13th centurynorth of France, a clever vertical version of this type of cyphers, of whomdesign was particularly well fitted for arithmetic operations, reaches theacademic mediums. Its particular structure not only permits operation upto 9999 but, thanks to its additive structure, made easy the four standardoperations. From
1 2 3 4 6
one obtains1+4=5 1+6=7 2+6=8 1+2+3=9
Each number is associated with a decimal value — unit, tenth, hundred,thousand — according to its position and symmetry along a vertical bar— i.e. :
11As a friend of Robert Grosseteste, chancellor of the University of Oxford and archdea-con of Leicester and later bishop of Lincoln, John of Basingstoke occupied a central posi-tion in the intellectual medium of the tenth century.
11
thousand hundred
tenth unit
as in5 3
94
= 5349
Even if no one has ever tried to compute with this ciphers, King (2001)has shown that they were perfectly apt to realize the four operations. Butthere was a major disfunction : it was impossible to define an algebra sinceif we define addition or multiplication as the internal operation there arenumbers not representable with this numeral systems12. Be that as it may,this type of cyphers hardly has been of any used outside one 14th centuryastrolabe of Berselius and it’s use for marking volumes on wine-barrels13.
Before the end of the Middle-age, not only the numeral notation wasdeficient, so it was for the technology of the operations, at least in Europa,was deficient. Multiplication and division was hard tasks operated onhands or with an abacus.
Of course, not later than the 5th century, indian mathematicians imag-ined a positional system with a peculiar use of the zero since both, thepositional system and the zero, were the basis of the sexagesimal system ofthe babylonian14. What make peculiar the indian numeral system is that,contrary to the 3th century b.c. babylonian numeral system, the zero wasnot just a mark to indicate a void position, but a true number.
The transmission through the arab mathematics to the west could onlysucceed in a civilization clash because, when the occident discovered thezero, it was under the monopoly of the christian doctrine which, at thattime was completely devoted to the aristotelician philosophy for whichthe void associated with the zero was truly a non-concept.
First of all, as underlined by Seife (2000), zero brake the rubber bandproperty of the integer numbers which come throughout multiplicationand associativity. That is to say that, for every number, with the exception
12In 1953, then at Bell Labs, Claude Shannon constructed a mechanical calculator whichcomputed in Roman numeral so as to demonstrate that, even if difficult, it was possibleto compute this way.
13In the middle of the 16th century, Gerolamo Cardano proposed his own version ofthe ciphers in his De subtilitate Libre XXI of 1550 — see Cardano (1999[1666]).
14One may wonder why babylonians used a sexagesimal system. One possible explana-tion is that with such a system 1/2, 1/3 and 1/6 are integers which simplified astronomicaloperations.
12
of zero, the multiplication by any integer change the scale, as if the rubberband expanded. Secondly, if one adds two numbers and multiply theresulting number by any number we obtain the same result as if one hasmultiply the two primitive numbers by the same number and then addsthe to resulting numbers. But with zero obeys its own rules : any numbermultiply by zero shrinks to zero and any number adds to zero stays thesame. Thirdly, as the pythagorician’s doctrine ascertained that every thingsthat make sense in the universe had to be related to a neat proportion zerocouldn’t fit in this doctrine as any number divided by zero is infinite, to thenotable exception of zero which, as is universally known, is undetermined.
So, to the notable exception of pythagoricians, if greek mathematicscould accepts irrational and negatives numbers, avare of the sumeriansexagesimal numeral system which, as one has seen earlier, has a peculiarzero, they could not accept zero as a number on philosophical basis, notby ignorance.
And then, Zeno of Elea sets out his famous paradox of Achile and thetortoise. If Achile raced the tortoise that has a head start, even if he runstwo time speeder than the lumbering tortoise, he will never catch up withthe tortoise since each time he made half the distance between his positionand the one of the tortoise that last one has increase the distance thatAchille must make15. Because of the lack of a the concept — greeks haveno word to name it —, it was impossible to cope with the paradoxe. Zenoconclusion was contrary to experiment : motion was impossible. Everything is one and changeless16. Here enter Aristote who in front of Zenoparadoxe declares that there is no need of infinity which are could existonly in the mind of mathematicians without actual support. There was noinfinity and no void, the earth was situated in the center of the universesurrounded by spheres in a pythagorian harmony. But as there were noinfinity, the number of spheres was necessary finite, the last of them beingthe celestial vault. There was no such thing as beyond the celestial vaultand the universe was contained in a nutshell.
15The solution will emerge from the development of mathematical analysis, with the in-troduction of the notion of limit by the indian mathematician Madhava of Sangamagramain the 15th century who was in all probability not aware of the problem, two centuriesbefore that Cauchy strictly develops the analysis of the subject on the basis of the seriesconvergence initialized by d’Alembert.
16More exactly, Zeno was a member of the eleatic school founded by Parmenides andhis paradox was designed to support Parmenides arguments.
13
The reason why this conception of the universe lasts so long is that itincorporated a proof of god existence. As there was movements17, some-thing must be causing the motion and, by necessity, it could be only theprime mover — i.e. : god. As expressed by Seife (2000), as wrong as itis, the aristotelician was so successful that for a millennium it eclipsed allopposing views.
It causes many problems the most part of them being linked to the cal-endar since it was impossible to define a year zero. As the catholic churchendorsed that philosophy, one should wait 1277 to see Etienne Tempier,bishop of Paris, condemn all doctrines contradicting god omnipotencewhich was the case for the aristotelician physics — see Piche (1999). All ofa sudden, void was allowed since an omnipotent deity has not to followany consequences of a human philosophy.
Unfortunatelly, Tempier condemnation was not the final blow of thearistotelician theory. The church will stay clinged to it for still some cen-turies, and even will reinvigorate this theory when attacked at the reformtime.
Because of all this sad history, in those times, only a very small bunch ofpeoples where able to go further that counting on the hand fingers18. Thenat that time, there has been a reversal in education. In the british islands,it stagnated. Of course, between the 12th and the 14th century, many greatuniversities were created in all Europe and the teaching of the mathematicsof the ancient rise — even new mathematics were developed and diffusedas the one of Fibonnaci19,20 whose best promotor were Johannes de Sac-
17As the earth was at the center of the world, it couldn’t move. So, the rotation of thecelestial on itself should be explained.
18In Turner (1951), one can find a table showing how roman peoples were using theirfingers to convey numbers from 1 to 99. More recently, Ifrah (1985[1981]) devotes acomplete chapter to the history of fingers counting.
19His Liber Abbaci, Practica Geometria has been written in 1202. The success was so greatthat he realized a second edition publish in his living in 1228. Up to now, all copies of the1202 have disappeared. For wha t concerns the 1228 edition only fourteen copies existsnowadays but only three, located in Italia, are complete or almost so and seven are merefragments. It should wait 800 years to be translated in English — see Fibonacci & Sigler(2003[1202]).
20In any case, was Fibonacci known under this name in his life time. In the tradition ofthat time, he would have been known as Leonardo of Pisa. But, in the introduction of LiberAbbaci, he refers to himself as filius Bonacci, a name which was not the name of his fatherWilliam — one knows also the name of a brother Bonaccingus, but nothing. Perhaps,
14
robosco — or John of HollyWood or John of Halifax — who learned andteach in Paris and wrote his Tractatus de Sphaera whose stayed one of themost studied astronomic book until the end of the Renaissance, the frenchAlexander of Villedieu’s Carmen of Algorism written in 1240 and JordanusNemorarius — still known as Jordanus of Nemore — Algorismus, a moreformal treatise, whose redaction date in the 13th century is unknown. Inthose days, the first translation in Latin of the Elements of Euclid was done,and at least in the 14th century, the mathematical curriculum included al-gorithmic, ptolemaic astronomy, perspective, proportion, measurement ofsurfaces and. . . fingers accounting which was a pre-requisite to the entryin the Universities.
What make Fibonnaci book so peculiar to his temporaries. The answeris astonishingly simple : it provided an original sum of knowledge linkedto the economic of trade which, up to that innovation hardly at disposaleven in the old technology of the roman numerals and, in a time whereilliteracy was the rule, were, at the end, easy to memorize and manage.
Not only it was the first occidental exposition of the hindu-arabic nu-merals which diffuses outside the friaries, but it gives also the algorithm tocompute the four operations and explains how to apply them to problemsessentially linked to the mathematics of trade, freeing the accountants ofany counting boards, contrary to what announced the title of the book since,until then, they were mandatory to compute any operation21. As such, he
was he proud to belong to the family of his mother. In 1838, the historian GuillaumeLibri take the freedom to contract filius Bonacci in Fibonacci. Occasionally, he refers tohimself as Bigollo which, in tuscan dialect signifies traveler and in others italian dialectsblockhead. It is under this name that Pise decide to grant to him a salary of 20 lira. In factthis is a very complex subject since it seems that, Perizolo, notary of the Roman Empire,mentions Leonardo as Fibonacci in 1506 — for more details on that subject, see Drozdyuk& Drozdyuk (2010).
21That Liber Abaci should have been the gate through which practical algebra has beentransmitted to Italy and then christian Europe has been convincingly recently disclaimedby Høyrup (2010). One can mention at least two early tentatives to work with hindu-arabic numeral : the first one was Rabi Abraham ben Meir Ibn Ezra who, in his 1148Sefer ha-Echad — Book on Unity —, borrowed the Indian place-value system, but insteadof using the traditional number signs, represented each with the first nine letters ofthe Hebrew alphabet — keeping the Indian sign for zero — and Thomas Le Brun yetknown also under the name Qaid Brun, an englishman who was in turn secretary ofWilliam I of Sicily ”The bad”, and later a reformer of the exchequer under Henry II, whoseems to have tried to introduced them in the usage of the exchequer but has ot beenfollowed near by 1158 — here the conditional comes from the fact that we have find some
15
was immediately adopted by merchants who, unexpectedly, could, all of asudden, acquire, in a time were knowledge was nearly a church monopoly,a modern technology which lowers the accounting costs.
For instance, even if it was not his own, Fibonacci introduced the mul-tiplication per jealousia which made this operation more accessible to themost part of peoples who were able to read. For instance, if one want tomultiply say 2345 by 467, as was exposed in the book, on can operate byassociativity in the same way as with roman numeral — i.e. : one can notethat 2345 = 2000 + 300 + 40 + 5 and 467 = 400 + 60 + 7.
× 2000 300 40 5
400 800000 120000 16000 2000 +60 120000 18000 2400 300 +
7 14000 2100 280 35 +
934000 140100 18680 2335 = 1095115+ + +
but we note that that method imply huge numbers multiplications. Thejalousia method22 of 9 by 9. This time too, one begin by a double-entrytable but in each cases one find the unit, the ten, the hundred, the thou-sand. . . then one operate as indicated in the following example.
recent sources on the subject attesting the existence of some documents which are notreferenced. According to Smith & Karpinski (1911), in the same period one can find thefirst computations with hindu-arabic numerals in a german Algorismus of 1143. Fibanaccihimself, in his introduction to Liber Abaci, tels us that, in what concerns hindu-arabicnumbers, while in business, he went in all the places where they were studied and taught— i.e. : Egypt, Syria, Greece, Sicily, and Provence — and, according to the translation ofGrimm (2002), he pursued [his] study in depth and learned the give-and take of disputation.
22According to the dictionary, the name comes from the jalousie window, a windowcompose of wooden louvers set in a frame. The louvers are locked together onto a track,so that they may be tilted open and shut in unison, to control airflow through the window.Jalousies are reputed to permit to people to see unnoticed.
16
ten
unit
×
121
1
2
08
12
14
3
12
18
21
4
16
24
28
5
20
30
35
4
6
7
5115901
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
In what concern the division, he introduced to the european public thegalley method which is obviously of chinese origin — see Lay-Yong (1966)23.
Suppose one wants to divide 7385 by 214. The method, which is heredecomposed stage by stage, began as shown here under :
①
7385 |214
As 2 ∙ 3 < 7 < 2 ∙ 4, one write 3 in the right place, write 2 under the 7and the rest in the difference between 7 and 2 ∙ 3 above the seven and dothe two same operations for the one crossing the numeral already used.
②
1��7385 | 3��214
③
10��7��385 | 3��2��14
Now 3 ∙ 4 = 12 but on the line there is only 8 so we must borrow a 1 to10. The remainder is 9 and the remainder of 18 -12 = 6 so
23This method stays nearly for three centuries the lone method taught, since it is papereconomizing and that the paper was first imported in occident in the 13th century.
17
④
∙9��1��02��7��3��85 | 3��2��1��4
Now, as we have exhausted the use of 214 we rewrite it shifted by oneposition.
⑤
∙9��1��02��7��3��85 | 3��2��1��44221
And one begins again.
⑤
∙1∙��9��1��06��7��3��85 | 34��2��1��442��21
⑥
∙��1∙��92��1��0��6��7��3��85 | 34��2��1��442��2��1
⑦
∙10∙��9��8��1��0��69��7��3��8��5 | 34��2��1��4��42��2��1
At the end of the process, one find that there is 34 times 214 in 7385 witha reminder of 109. This method generate at least two remarks: first, it is notself-evident for a learner to understand that a number couldn’t be writtenon the same line, but this is marginal. Secondly, there was no attempt to tryto make any expansion of the remainder in decimal value as one uses to donowadays. It was not in the spirit of the time and, even if it has been knowin China, in Persia long ago before Fibonacci, even if it was developed inEurope in the 14th century writings of Immanuel ben Jacob Bonfils, it willbe necessary to wait the 17th century and the De Thiende of the flemishmathematician Simon Stevin24 so that the decimal development becomes
24To help the diffusion of his thinking, Steven had translated his work by himself.
18
an universal practice25 — see Stevin (1935[1585]). But, from a practicalpoint of view, that was perfectly justified. In a tradition inherited from theegyptian, it was the usage to write a non integer number as A.bcd in theform :
d10
c10
b10
A
where the denominator of each fraction was understood as the product ofall denominator value that preceded it26. For instance, Fibonacci wouldhave written the result of the former division 7385 ÷ 214 as 109
21434. Supposenow that this value be a monetary value. In his living time, the subdivisionof monetary unit was complex27 but with the Parma system of accountFibonacci would have written 2
206
1234 that is 34 Lira 6/12 of Lira and 2/240of Lira or in actual money 34 Lira 6 Soldi and 2 Denari.
An other innovation of the Liber Abaci, is it advocacy for the use ofnegative numbers which Fibonacci interprets as debts or, in another work— Flos28 —, throughout the deficit metaphor. Of course, even in that cas,Fibonacci was just a diffuser since negative numbers were of chinese and,later, indian origin. And even his interpretation was not original since,in the 7th century, the indian mathematician Brahmagupta has yet calledthem debts :
A debt cut off from nothingness becomes a credit;a credit cut off from nothingness becomes a debt.
The true innovations of the Liber Abacci could be find in its numerousapplications and nearly all applications, one can find in that book, are of
25But, one more time, we know now that in the middle of the 10th century, al-Uqlidisiof Damascus introduced place-value decimals to the right of the decimal point. Unfor-tunately, non one saw a particular reason to adopt it and this brillant idea sleeps for fivehundred years before that arab scholars awake it and tree more centuries before Stevenconvinces europeans of its power — see Devlin (2010).
26It is clear that this way to write a number is of arabic origin since, under this structure,it is clearly written from left to right. One must also note that the fraction bar is a Fibonacciinnovation.
27Until six subdivision as Ducado, Lira, Grosso, Soldo, Piciolo and Denaro, but there wasno uniformity between towns. For instance, in Parma, the main money was the Lira,which was subdivided in 20 Soldi themselves subdivided in 12 Denari. It is notable thatunit of weight and measure were even more complex.
28See Fibonacci (2010).
19
commercial and financial content. That is not to say that applications werenot developed as a by product of the development of the Algebra andessentially from the work of Muhammad ibn Musa Al Khwarizmi fromwhich Fibonacci transmitted essentially all the mathematical innovationsbut, the main difference is linked to the fact that, apart of his involve-ment in geometry, Khwarizmi’s exemples are all essentially linked to legalproblems of inheritance. Without any means to know that by himself,there are evidences, through the identity of some developed examples,that Fibonacci commercial and financial investigations are linked to thoseof earlier indian mathematicians as Aryabhata who, stradding on the 5thand the 6th century, developed some interest calculi in his astronomicalbook Aryabhatiya — see Clark (1930) —, Bhaskara whose commentarieson Aryabhatiya, written in the 7th century, include some problems relatedto partnership share divisions, and the relative pricing of commodities —Goetzmann (2004) — and Sridharacaryas who, in the 10th century, devotesome of its 300 verse couplets Trisastika, to a few practical interest rateproblems and a division of partnership problem — see Ramanujacharia &Kaye (1913). In Liber Abacci , one can even find a 9th century copy & pastproblem borrowed to the jain mathematician Mahavira’s book, Ganita SaraSangraha, where three merchants find a purse lying in the road. The firstasserts that the discovery would make him twice as wealthy as the othertwo combined. The second claims his wealth would triple if he kept thepurse, and the third claims his wealth would increase five fold29.
But, two innovations denote a true financial innovative spirit. First, inLiber Abacci, one can find the first calculus of an equilibrium price as theresult of absence of arbitrage.
Try to imagine how to determine an unknown price from a given quan-tity of merchandise when the price per unit is known30 — suppose a 100rolls costs 40 lira, how much would five rolls cost ? — without the apparatus
29In fact, nearly all, what in a near past, appeared to be Fibonacci innovations havebeen demonstrated borrowed to indian or arabic traditions. For instance, the celebratedFibonacci sequence of the growth of a rabbit population has been borrowed to the Chan-dahshastra — The Art of Prosody — of the sanskrit grammarian Pingala, who was activesomewhere between 450 and 200 BCE. Then, the indian mathematician Virahanka showedhow the sequences arises in the analysis of metres with long and short syllabes. Finally,around 1150, the jain philosopher Hemachandra composed a text on the term value ofthis numbers — see Devlin (2010).
30It’s the subject of the chapter 8.
20
links to the resolution of a first order equation with one unknown.Nowadays, one would write :
x ∙ 100 = 40 =⇒ x =40100
=⇒ x ∙ 5 =40100∙ 5 = 2
a solution that, nowadays, nobody can find remarquable since first orderequations are taught in the beginning of the secondary school. Leonardopresent it under a different structure as the famous rule of three — i.e. :
40 100? 5
4 ∙ 520
(=?)
As remarked by Goetzmann (2004), the rule of three is one of the oldestalgebraic tools since it appears for the first time in the Aryabhatiya, andis extended and elaborated upon in Bhaskaras commentaries, in whichhe applies it to problems quite similar to those analyzed by Leonardo— see Sarma (2002). It’s so simple that we can hardly imagine how wereperform exchanges before it came of common knowledge by every traders.Of all evidence, in every transaction, one of the two party was certainlydefrauded. As a proof for that case, one can advance that, if it has not beenthe case, Fibonacci should not have developed so many complex exampleson that subject. Immediately, Fibonacci explains how to find the price ratiobetween two goods
It is proposed that 7 rolls of pepper are worth 4 bezants and 9 pounds of saffronare worth 11 bezants, and it is sought how much saffron will be had for 23rolls of pepper
One more time for the common run of people living in the XXIth mil-lennium, there shouldn’t be any difficulty with this operation : One findthe unit price of the roll of pepper, the unit price for the pound of safran,then one calculate the price ratio and, after multiplication by the numberof desired rolls, the mass is said, that is :
7 ∙ pp = 49 ∙ ps = 11
}
=⇒pp
ps=
4 ∙ 911 ∙ 7
that is to say that if one wants to exchange 23 rolls of pepper against saffronone will obtain :
21
( 4 ∙ 911 ∙ 7
)∙ 23 =
211
87
10 ≈ 10.75324675 . . .
But, on more time it’s only the application of the rule of 5, that Bhaskaraand later Indian mathematicians developed for expressing price/quantityrelationships across several goods — i.e. :
Saffron Bezants Pepper
? 4 79 11 23
Implicitely, Fibonacci teaches how, in that matrix, the balance is betweenthe two diagonals
Saffron Bezants Pepper
? 4 7
9 11 23
and
Saffron Bezants Pepper
? 4 7
9 11 23
and finally give the thing — ?.Up to this point, Fibonacci also shows how to arbitrate between moneys
because, in that time, even in Tuscany, even if cities have inherited of thelate roman money division — denari, soldi, lire31 —, the relative value andmetallic composition of the various moneys varied considerably throughtime and across space32. On that subject, after 1252, the teachings of theLiber Abacci, will became unavoidable, due to the introduction of the goldenflorin which was the first money which could simultaneously serve as aunit of account and transaction on a 1:1 basis, since there were evidence,that it was never debased — see Velde (2000).
But, as bring out by Goetzmann (2004), and echoed by Rubinstein(2006), it’s in the finance field that Liber Abacci shows all his genius indiscussing four type of problems :
① How to split fairly the profit of a joint venture when contributionsare unequal, are made at different points in time, and in differentcurrencies or goods and in cases in which business partners borrowfrom each other
31Which finally remains in the english pounds, shilling and pence system.32It’s the reason why, he also give many exemples of minting and alloying of money.
22
② How to calculate the profits generated by a sequence of business tripsin which profit and expense or withdrawal of capital occurs at eachstop.
③ How to calculate the future values of investments made with bankinghouses.
④ The first use of present value analysis as a criterium to evaluateinvestments, including specifically the difference between annualand quarterly compound interest.
In what concerns the first point, on must know that in the 13th centuryitalia, the business ventures were organized through the commenda con-tract33 which stated how the commender — the partner who invested hisfunds — and the traveler — the partner who invested his labor — shoulddivide the profits. If this last one, doesn’t abound to the financement, thecommenda was unilateral and the commender retained 3/4 of the benefits.If the traveller decide to abound to the financement, in which case thecommenda contract was reputed bilateral, it was generally up to half of thecommender abounding.
Then, according to the specific dispositions of the contract, the studyof notarial archives from cities as Barcelona, Genoa, Venice, Amalfi, Mar-seilles, and Pisa34 reveals that, generally, any profit was usually divided1/2-1/2 while the commendator bore 2/3 of any loss and the tractator 1/3.But variations of this dispositions could be encountered. For instance, inDubrovnik, the share 3/4-1/4 of the unilateral commenda was not a rule atall.
Here, Fibonacci innovates in explaining how to fairly — that is accord-ing to the initial contribution — divide the profit when there is more thanone commender — in those time, they constitute a societas35.
In what concern the travelling merchant, here is how Fibonacci stateshis basic problem :
33Prior to Pryor (1977), it was common to place the origin of this contract in the muslimqirad contract. Since then, the Jewish ’isqa contract and the roman societas appear to betwo other sources of the commenda contract.
34According to Fibonacci & Sigler (2003[1202]), the Constitutum Usus of 1156 is the theearliest surviving municipal document specifying the conditions of the commenda contract.
35For the specific operations, one can consults one more time Goetzmann (2004) orFibonacci & Sigler (2003[1202]).
23
A certain man proceeded to Lucca on business to make a profit doubledhis money, and he spent there 12 denari. He then left and wentthrough Florence; he there doubled his money, and spent 12 denari.Then he returned to Pisa, doubled his money and it is proposed thathe had nothing left. It is sought how much he had at the beginning.
Here, one can find the solution original under the restreint that wordshave the same meaning for us than for Fibonacci, that is to say that onemust understand that, in returning in Pisa the man spent also 12 denariand that the money double before the expense.
Here is how it may be reasonable to solve the problem by a forwardargument. If X is the initial capital, one has for an initial capital of x, acapital of 2 ∙ (x− 12) in Lucca, of 2 ∙ ((2 ∙ x− 24)− 12) = 4 ∙ x− 36 in Florenceand of 2 ∙ ((4 ∙ x − 36) − 12) = 8 ∙ x − 84 at his return in Pisa, which is finallyequal to 0, process which give birth to the forward dynamic scheme
StandardSolution
FibonacciSolutionPisa
x
8 ∙ x − 84(= 0)
Luca
2 ∙ x − 12
Florence
4 ∙ x − 36
Pisa
122 + 12
4 + 128 = x
12
Luca
12 + 122 + 12
4
Florence
12 + 122
As finally, he has spent all his wealth, one must solve the first orderequation 8 ∙ x − 84 = 0 which give x = 10.5. This is how Fibonacci couldhave simply solve his problem.
But as the sign of a great spirit, he find the solution in following a com-plete new path, which could certainly be identified as the first backwardargument in the history of thought: he actualizes.
He states that, since its capital double at each stop, its initial capitalis 1/2 the capital he owns in Lucca, which is 1/4 the capital he owns inFlorence, which is 1/8 the capital he owns at is return in Pisa. In terms ofthe initial capital. That is to say that a denaro spent in his third stay couldnot be evaluated on the same basis as a dinaro spent in his second and in
24
his first stay. So, 12 dinari spent in his final stay is worth only 1/8 of 12dinari owned at the departure of the trip, 12 dinari spent in is penultimatestay ar worth 1/4 of the dinary owned at the departure and, 12 dinari spentin his first stay are worth 1/2 of the initial dinari. That is to say that, onecould evaluate a flux of dinari at different dates at the departure time. Thisgive :
Total Expense =122
+124
+128
= 10.5 = Initial Capital
One must also remark that, at least by implication, Fibonacci’s solu-tion is linked to an anticipatory conscience of double entry bookkeepingsince his solution discriminates clearly between capital and expenses inthe sequences of accounts
Pisa
Ressources Expenses
x
Lucca
Ressources Expenses
2 ∙ x 12
2 ∙ x − 12
Florence
Ressources Expenses
2 ∙ (2 ∙ x − 12) 12
4 ∙ x − 36
Pisa
Ressources Expenses
2 ∙ (4 ∙ x − 36) 12
8 ∙ x − 84
One must note that as simple that those accounts may seem, it was aformidable achievement to write it, since the successives ressources incor-porate negative numbers which he has just introduced in an earlier chapter.Now, in an exercice of experimental archeology one can try to reconstructhis argumentation. As one can interpret negative ressource values as van-ished earning opportunities, they must slither from the Asset column tobe written in the Liabilities column as shown hereunder.
Pisa
Assets Liabilities
x
Lucca
Assets Liabilities
2 ∙ x 12
2 ∙ x − 12
Florence
Assets liabilities
4 ∙ x Lucca : 2 ∙ 12
Pisa : 12
4 ∙ x 36
Pisa
Assets Liabilities
8 ∙ x Lucca : 4 ∙ 12
Florence : 2 ∙ 12
Pisa : 12
8 ∙ x 84
Under this presentation, it is self-evident that the last Pisa’s book iswritten in terms of accumulated assets, not in terms of initial assets. SoFibonacci must simply have divided all terms by 8 to finally obtain thereturn to Pisa book in terms of initial assets, that is :
25
Pisa
Assets Liabilities
x Lucca : 12 ∙ 12
Florence : 14 ∙ 12
Pisa : 18 ∙ 12
x 10.5
What a tour de force ! For the first time, since human beings triedto maintain the balance sequences of his commercial operations realizedin distinct places, Fibonacci, from scratch, shows how to relativize all theentries and express them in initial value entitling the comparison of variouscomplex flux of income. And more than that, to realize this tour de force, hismind, at least tacitely, should have been aware of the possibility to keepall the entries by the distinction between income and expenses — i.e.: Adouble-entry bookkeeping operations which, as one will explain later inthis paper, will be truly available only two centuries later.
There is no evidence to state if Fibonacci has derived is invention ofactualization, but it is more than a conjecture as can be seen from theproblems which follow the travelling merchant problem. They are a set ofsophisticated banking problems such as :
A man placed 100 pounds at a certain [banking] house for 4 denari per poundper month interest and he took back each year a payment of 30 pounds. Onemust compute in each year the 30 pounds reduction of capital and the profiton the said 30 pounds. It is sought how many years, months, days and hourshe will hold money in the house.
It is worth the effort to follow, helped by the modern apparatus, theeffort of Fibonacci to find the solution of his problem. According to thetraveller’s one, one must reason as follow. First of all, one must find theinterest rate and, according to the decomposition denari, soldi, lire, as a lire— a pound — is worth 240 denari, and the return of the investment is 400denari per month for 12 month, that is to say 4800 denari or 20 lira/year,the rate of return is .2. Now, one has :
26
0 100 −301.2
+30
1.22+
301.23
+30
1.24+
301.25
+30
1.26= 0.234697
1 1.2 ∙ 100 − 30 = 90
2 1.2 ∙ 90 − 30 = 78
3 1.2 ∙ 78 − 30 = 63.6
4 1.2 ∙ 63.6 − 30 = 46.32
5 1.2 ∙ 46.32 − 30 = 25.584
6 1.2 ∙ 25.584 − 30 = 0.7008
So there is a balance. And, as this balance is lower than 30, one knowsthat the account cannot stay open one more year. So one must reason indays because month are not regular unit. But to reason in days, one mustalso suppose that, all things equal, all the magnitudes are time homoge-neous, that is to say that the man’s daily expanses are equal to the 1/365thof his yearly expenses and that the daily interest rate is also simply 1/365thof the yearly one.
This carries to the new table :
0 0.708 −
( 30365
)
(1 +
2365
) +
( 30365
)
(1 +
2365
)2+
( 30365
)
(1 +
2365
)3+
( 30365
)
(1 +
2365
)4+
( 30365
)
(1 +
2365
)5+
( 30365
)
(1 +
2365
)6+
( 30365
)
(1 +
2365
)7+
( 30365
)
(1 +
2365
)8= 0.641613
1(1 +
2365
)∙ 0.708 −
30365
= 0.629688
2(1 +
2365
)∙ 0.629688 −
30365
= 0.550946
3(1 +
2365
)∙ 0.550946 −
30365
= 0.471773
4(1 +
2365
)∙ 0.471773 −
30365
= 0.392167
5(1 +
2365
)∙ 0.392167 −
30365
= 0.312124
6(1 +
2365
)∙ 0.312124 −
30365
= 0.231642
7(1 +
2365
)∙ 0.231642 −
30365
= 0.15072
8(1 +
2365
)∙ 0.15072 −
30365
= 0.0693538
27
So, one has found that after 6 years and 8 hours the account will benearly void — Fibonacci goes further since he displays the exact answerwich is 6 years, 8 days and 1
2395 hours in his own notation but one renounces
to go to this stage not to tire the reader. But what a fantastic operation.It outperform any performed computation before centuries. And fromthis problem, Fibonacci constructed 11 other examples from which thefollowing has been extracted because it’s has been viewed as the foundingproblem of the modern finance since for the first time in history, not only thepresent value criterium is applied to discriminate between two apparentlyidentical payment which differs by their sequence. It’s the celebrated On ssoldier receiving three hundred bezants for his fief.
The Fibonacci’s story invented to expose this sui generis problem is thestory of a soldier of whom the King want to reward his service record ingranting him an annuity of 300 bezants/year, paid in quarterly installmentsof 75 bezants. Then the King alters the payment schedule to an annual year-end payment of 300 bezants. Knowing that the soldier is able to earn 2bezants for one hundred invested bezants, Fibonacci asked if the situationof the soldier is better in the first or in the second schedule.
It’s clearly a simple present value problem which can be analyzed inthe first year of its payment. In the case where the grant is served in oneshot, as the mensual rate of return that is accessible to the soldier est of 2%,the present value is :
V300 =300
1.0212= 236.548
and in the case of the quarterly payment of 75 bezants, one find :
V75 = 75 +75
1.024+
751.028
+75
1.0212= 267.437
So after alteration of the schedule, it comes that, in present value, thepension has been altered 30.8892 of bezants: The soldier should have beenbetter off, if the king shouldn’t have altered the payment schedule. As a byproduct of this accomplishment, and only as a by product, Fibonacci devel-oped the rabbit population growth problem and as a natural consequence,the first analysis of a geometric sequence of number.
It could be tempting to minimize the stupendous financial accomplish-ment for Fibonacci pretending there is no news under the sun since capi-talisation is an operation which was used before the Hammurabi code that
28
is 18th b.c.. It’s a very simple operation which amount to compound thedue interests at the expiration date. And, as shown by Goetzmann (2004),one can also be astonished by the incredible connection between the LiberAbacci and the babylonian problem exposed in the tablet 8528 conservedin the Berlin museum and published by Neugebauer (1935).
If I lent one mina of silver at the rateof 12 shekels (a shekel is equal to 1/60of a mina) per year, and I received inrepayment, one talent (60 minas) and4 minas. How long did the money ac-cumulate?
tablet 8528 Liber Abacci
A certain man gave one denaro at inter-est so that in five years he must receivedouble the denari, and in another fivehe must have double two of the denariand thus forever from 5 to 5 years thecapital and interest are doubled. It issought how many denari from this onedenaro he must have in 100 years.
If capitalization is necessarily a by-product of the loan operation, asdemonstrated by the nearly three thousand rolled years between the firstoperation of capitalisation and the actualization by Fibonacci, that lastoperation is linked to far more complex mental schemes.
But, as necessity has a value of law, this great innovation couldn’t staya dead letter. And one knows that it has not been the case. CertainlyFibonacci was a first class teacher who never spared himself in diffusingthe new way to calculate without a mechanical support36 or to promotehis new financial instruments. One can support this assertion by threeremarks: First, during his life time, precisely in 1228, there has been asecond edition of the book, which was an exceptional event for such a book,secondly his fame arrived early to the ears of the Fredrick II37, emperor ofthe Holy Roman Empire, whose court mathematician — John of Palermo —
36It would be tempting to write here with a pencil and a sheet of paper, but this wouldn’trepresent Fibonacci’s actuality.
37It seems that he has read by himself Fibonacci’s book.
29
was charged to ask three problems to him38, two of them been either laterexpended in his Liber Quadratorum — see Fibonacci & Sigler (1987) — orincorporated in his Flos — currently untranslated, at least from the latintranscription of Fibonacci & Boncompagni (1857-1862) — both publishedin 1225. Third, in his old age, in 1241, Fibonacci receive a pension ofPisa which acknowledged equally his effort in the two distinct field of theeduction of the citizens and dedicated service39. That is to say that, duringhis time life, Fibonacci helped his town in performing fructuous financialoperations.
But the effort of a lone poor mathematician to transmit a new way to copewith number would have been a waste, if it shouldn’t have been relayedby some other means because, first the editing technology of that time wasso expensive that only the happy few could have access to books.
A book — initially named a Codex — was a technological progress onthe Volumen — a scroll of papyrus — because it was far more durable thanthe later. It was design on the same structure that the assembly of woodenand wax tablets used for drafts and provisional texts. But they were also farmore costly since the pages were made of parchment, that is to say sheephide, even if parchment were apt to support reverse writings, support tobe fold and sewed in registers.
But, more than that, books were hand written and, since this wasvery costly time consuming operation, extraordinary costly. For example,Dittmar (2011) yield that in 1383, in England, a scribe was commissionedto write a single service book for the bishop of Westminster. For this work,the scribe was was paid £4, a sum equivalent to 208 days’ wages for a
38The first problem is a problem looking like those resolved in Liber Abaci. The secondwas to find a rational number r such that both r2 − 5and r2 + 5 are rational squares forwhich, Fibonacci established that if r2 − n and r2 + n are rational squares, then there isa right triangle with rational sides, hypotenuse 2 ∙ r , and area n, which conduct to thesolution n = 5. In what concern the third problem — find a root of the cubic equationx3 + 2 ∙ x2 + 10 ∙ x = 20 —, according to Brown & Brunson (2008), it was borrowed to OmarKhayyams Al-jabr — see Kasir (1931-1972). Fibonacci proved that the solution of thiscubic equation was neither an integer, nor a rational, nor a number of any of the formsfrom Book X of Euclid’s Elements. So he decide to approximate it. Unfortunately, he failedto give a good approximation since the number he gives was neither a truncation nor arounding up of the actual root — for more on that subject see Brown & Brunson (2008)who try to explained why he made this mistake.
39See H. Lhuneburg post at http://www.mathematik.unikl.de/˜luene/miszellen/Fibonacci.html
30
skilled craftsman knowing that the costs of illustration, binding, and pa-per were listed separately and that he performed the transcription parttime and completed the project over the course of two years. During themanuscript era, books were sufficiently rare and valuable that they wereused as collateral on substantial cash loans extended to scholars by Oxfordand Cambridge universities — Bell (1937).
Initially produced in the scriptoria of the friary, the most part of thecodex was devoted to religious books. But, in the 12th-13th century, slowly,things began to change. In the secular world, the extraction of the scholareducation from the cathedral and monastic schools to the Universities fromBologna (1088), to Siena (1240), while passing through La Sorbone (1150),Oxford (1167), Palencia (1208), Cambridge (1209), Salamanca (1218), Padua(1218), Montpellier (1220), Naples (1224) and Toulouse (1229), created andexpanded markets for law, mathematical, medical. . . book began to de-velop due to a small literacy increase, causing the opening of scriptoriain towns but this movement stayed marginal — less that 25 % of themanuscript of this time are devoted to non religious subjects. So, a booklike Liber Abaci could be written only for a very small audience which wasyet ready to receive his message because, to accept to be involved in sucha purchase one must have been persuaded that it was worth the expense.So the book was written necessarily for rich merchant and more preciselyfor those who were called to rule the city.
In what concern the replacement of the roman numerals by the hindu-arabic numerals, the audience was certainly not too difficult to convince,because with the development of exchanges with the arabic world, it wentto be secure that the later outshine the former. After all, as recorded byFibonacci himself, his father, who has been send in Bugia40 to supervisethe pisean commerce with this tunisian town, was persuaded that his soncould, to some purpose, find some interest in learning the arabian wayof computing. There is no reason to doubt that he was alone of his kind,even if he was the lone to have a son which became able to convert it in ascientific revolution.
40Now Bejaia in Algeria. In those times, Bugia was one of the main intellectual centers ofthe arabic world of equal fame with Sevilla and Toledo. Strikingly, the last crusade whichended under the walls of Tunis after the passing of Louis IX of France by a treatise betweenCharles d’Anjou and Tunis’s emir which bestowed, among other things, to christian theright to free trade in the Tunisia territory. But in that time, Fibonacci was dead since 40years.
31
Secondly, toscan merchants involved with mediterranean trade werecertainly already ready to cope with the hindu-arabic numeral. Because,if from 1095 to 1291, the crusades failed to hit their main objective, whichwas, in the time terminology, to free Jerusalem of the faithless, despitethe recurrent conflicts which will kept on going between christians andmuslims, they will have, as unintended consequences, to secure the wealthof merchant italian cities. Their wealth began by the provision of ships totransport many of the crusaders to the Middle East and set up with thetrade perfumes, spices as saffron, jewels, silk, dyes, tapestries, ivory andother products which the Europeans gradually came to value41.
The croisades influenced also the intellectual development of Europe.Above all they liberalized the minds of the crusaders which were in con-tact with the leading science from which they had so much to learn. Inparticular, certain great spirits began to untie from the aristotelian tradi-tion endorsed by the catholic church. One can easily conjecture that, inthat manner, the ground was set for merchants to endorse hindu-arabicnumerals.
And Fibonacci has been capable to convince his pisean fellow citizens,the inhabitants of the other main toscan and venetian cities and, finally, theworld. But this will take more than a century and at the end Pisa wouldhave been absorbed in the florencean contado42. The more astonishing isthat, at least for a contemporaneous literate man, Liber Abacci was a bigand difficult book whose messages could only be assimilated by peoplesready to set aside the ancient way of doing calculation
The instruments of the diffusion of the Fibonacci’s canons, were multi-ples : in a first time, the Liber Abaci litigated for himself toward the educatedpeople and after the approval of the Emperor in 1228, the official start wasgiven to try to transmit the new corpus to merchants.
Apparently, Leonardo was the first to teach and practice his methodsince, according to the State archives of Pisa for 1233-1241 — see Bonaini(1858) —, among other reasons adduced to the payment by the city of a 20lira pension, one specifies that he taught the new method with discretionand wisdom.
41On that subject and, more precisely, on the role and luck of Pisa in the mediterraneantrade, one could consult Tangheroni (2002) and Tangheroni (2003).
42During the Middle-Ages, in the north of Italy, the contado was the territory undercontrol and dependant of a major city. Pisa, short of its maritime power, will loose itsindependence in 1406.
32
If he should have act alone, there would have been no revolution be-cause, in those times, education was restricted to a minimum which washumanities skewed, and whose access was reserved to a few number ofpeoples. At the primary level, even in Italy, one can say that no things thatreally matter have changed since, in 789, Charlemagne has enacted thatevery cathedral and monastery should open schools, reserved to a maleassistance, where they could learn grammar, rhetoric, logic, latin, astron-omy, philosophy and mathematics — essentially arithmetic and geometry.Essentially, young peoples learned to read and write in latin et vernacularlanguages — see Durant (1950).
By slow degrees, and for at least four centuries, the first system of com-mercial private and/or public schools was organized in Tuscany. Accordingto Ulivi (2002), as soon as 1265, a man named Pietro, who should have beena direct pupil of the maestro, appears as witness in a bill of sale. This Pietrohad a son and both of them were maestro d’abaco. They certainly taught inprivate school. During the last quarter of the 13th century, one can testifythe existence of some communal abaco schools, the oldest testimony com-ing from the municipality of San Geminiano who hired a Michele in 1279.In 1282, Bologne institutes its own school. In the first rotuli of BolognaUniversity, in 1384,1388 and 1407, one find that a Antonio Bonini Biliotti,a university professor, should lecture in arithmetics, and geometry at thepre-universitary level, a decision which will stay in vitality at least untilthe middle of the 16th century. In the 15th century, Scipione del Ferro, whowas the first to use the imaginary numbers43, would be one of the Maestrodel Abaco. A similar situation appears to have been set in Perugia where in1389 and 1396, it is attested that, near by the secondary school professorof grammar, there was a professor of geometry and abaco. Interestingly,
43According to Cardan, del Ferro find the universal solution to second order equations.But et never published it since, in that time the university mathematical pulpit wasregularly put in competition, according to a rituel where the resolution of arithmeticproblems was mendatory. Dying, he bequests his solution to his son-in-law, Hannibal deNave, and one of his students Antonio Fior. This last one in competition for a universityvacancy with Nicolo Fontana — Tartaglia that is to say the stammerer — was unwise inasking this alter one to resolve too much third order equations. Tartaglia guessed theexistence of this universal solution and finally found it. Then Cardan, called to care theagonising Tartaglia, convinced him to transmit to him his knowledge. Tartaglia acceptedunder the strict condition that Cardan kept the solution for himself. Fortunately for us,Cardan doesn’t respect his parole, but beeing one of the best physician of his time hereally cure Tartaglia who recover his health and the controversy began.
33
as it can be deduced from Ulivi (2002), the municipality had deliberated afirst time in 1277 on the opportunity to create a school but the decision wasgained only after the third deliberation in 1285. During their life time, theVerona schools will see the teaching of Tartaglia and of Francesco Felicianode Lazisio both in the 16th century. It’s only in 1399 that Pisa hired twoprofessor for the creation of a local school.
One must also notice that small commercial towns picked out a pub-lic school system contrary to the great ones, as Florence or Venezia whichopted for a private one. According to Black (2004), Florence never followedthe practice of other toscan towns which propounded academic subven-tions to their citizens. This certainly explain the differential treatment ofthe professors which was dependent of the fortune of the cities where theytaught.
In Venezia, one dispose of testimonies that atteste the presence of AbacoSchools since 1305 and, as time goes by, the number of students of thoseschools could reach a little less of 150. The most famous one was theScuola di Rialto, established in 1408, where instruction covered logic, naturalphilosophy, theology, astronomy and mathematic, and the teaching wasof such good level that it was a kind of foundation courses for being acandidate to the University. And its least title of glory was to have hadLuca Pacioli as pupil. Around 1360-62, Leonardo da Vinci received aformation in his birth town just before being send in Florence to enter inAndrea Verrochio’s studio. If one doesn’t know with precision the birthdaydate of Piero della Francesca, one knows that, before 1430, he was a pupilof a scuola del abaco.
According to the memories of his father Bernardo, in 1480-1481, NiccoloMachiavelli studied also in such a school — see Machiavelli (1954[v 1488]).
In what concern Florence, for which we have testimonies since 1283,the reputation of the dispense teaching was so high that the city trainedthe most part of the maestri del Abaco who teached in Toscany and Venetiafor at least tree centuries. Between, the end of the 13th century and the firstthird of the 16th century, nearly sixty maestri have been counted whichoperated in twenty schools44. For the world education standards of thosetimes, Toscany appears truly outstanding. According to the chroniclerGiovani Villani — see Villani (1906[1341]) —, quoted in Davis (1965), that
44Ulivi (2002) gives a very elaborate description of the implantation of the schools inthe diverse districts of the town.
34
approximatively 10 % of Florence’s population, that is to say between 8000and 10000 boys and girls were being taught to read. One fourth of theboys — between 1000 and 1200 — decide to study in one of the six scuoladel abaca45. The social basis of recruitment was broad from patriciate tosmal shopkeepers. And, as noted by Goldthwaite (1972), they learned theirlessons well :
anyone who is familiar with the complexity of the arithmetic problems oftheir monetary systems, international exchange rates and other accountingand business practices and who has tested the accuracy of the calculations intheir commercial recors can vouch the effectiveness of the training Florentinesreceived in scuola d’abbaco.
One must also note that the black plague which, in 1348, spread fromGenoa to Rome invading Toscany, ravaging Florence to finally reach thepapal states, only slowed down the development of the merchant educa-tional system46. On the content, duration and organization of the studies inthe scuoli, we are pretty well documented, but one throw back to Goldth-waite (1972), Ulivi (2002) and Black (2007). One must shortly underlinethat those schools were perceived as high school accessible with the pre-requisite of the attendance to a primary cycle where children learned toread and to write. In average, the school days began at 10 or 11 years andlast two days but according to the competence and abilities of the child itcould variegate accordingly.
After the school, they enter as novice in the trade but not only becauseswiftly others corporations find that the formation dispensed in the scuolicould represent a good prequisit. Even if geometry had a very small part inthe teaching of the scuoli, this part was sufficient and largely outperformwhat was taught on that subject in other places all over Europa. Forinstance, Zervas (1975) argues convincingly that the design of the florentinebaptistery door of Santa Maria del Fiore by Andrea Pisano owe much tothe abaco teachings.
45The others, either came in one of the fourth latin school to study grammar and logic— nearly one eight —, in rethoric school, other in canon law school — certainly thecathedral school, and the last part, was taught in notarial schools. This crumbling of theformation was certainly linked to the fact that, in 1321, Florence was fighting to found auniversity.
46During the black plague the most part of the great cities had lost more than 40 % oftheir inhabitants. Florence lost more of the third of its population in the first six month,and between 45 % and 75 % in the first year.
35
Nevertheless for a life expectancy of less than 25 years to enter in thetrade at the age of thirteen years after two years of formations was themost than can be demanded in those times.
To help them in their teachings, some of the maestri departed fromFibonacci and had draft new books.
As more and more clergymen became mathematical educated, helpedby the fact that Dominicans, who were more involved in education thanolder orders, attracted more and more poor young peoples eager to be-come monks, the mathematical education raised at all levels even if theused methods of teaching conducts more to rote learning than true under-standing47.
But, during all those times, the far most advanced mathematical teach-ing was done by the trade guilds. The apprenticeship lasted seven years :a master of a trade then taught to an apprentice all he thought he shouldknow.
Of course, those studies were purely practical and no apprentice couldhave learned more that what artisans and merchants could teach classeshad he wanted to. Architect and builders but also merchants and tradersand in the big cities the early forms of money landers use to learn geometryand arithmetics.
So from the trades, because they imposed the study of mathematics totheir practitioners, came the conditions for a rapid advance in accountingtechnology which will be achieved during the Renaissance.
We must underline also that, if in 976 the Codex Albelendsis seu Vigilanuswas the first document to use arabic numeral, it take nearly 300 yearsbefore Leonardo Fibonacci, in his Liber Abbaci, advocated their universaluse in replacement of the roman numerals48. But, and this could explainwhy trades used them so early, the spreading of their use come also asa by product of teh development of the oriental trades generated by thecrusades and perhaps from the play of cards, whose origin is uncertain butlikely situated in orient or in the middle-east. In the primitive literatureon the subject, it is postulated that they were introduced in Europe by thearabs.
47Pupils were obliged to learn the question they could ask to the teacher and to learnalso the answers.
48The importance of the Fibonacci advocacy was early recognized by the italian cities.As a proof, he receives a permanent income from Pisa, certainly to teach arithmetics.
36
From playing card with arabic numeral to bookkeeping with the samemedium the step was natural for merchants who rapidly could not usetwo counting device, and the rise of the commercial relation of cities likeVenezia or Genova was a good incentive to develop a new accountingsystem.
This new accounting system was finally explained and advocated byLuca Pacioli whose fame is shown in its enigmatic portrait49. As a Re-naissance man, Pacioli accepted the interrelatedness of all the subject un-der study in his time : religion, business, military science, mathematics,medicine, art, music, law, language. He finally acquired a high level ofknowledge in all those fields but he cherished the most those which exhib-ited harmony and balance as mathematics and accounting — see Alexander(2002).
If Pacioli is often credited to be the father of double entry bookkeeping,he never claimed for himself its invention. The most part of the authorswho write on this subject argued that double entry bookkeeping was acommon practice since the 13th century italian cities but there are rareexceptions as Kats (1930) who convincingly argued that it was a commonpractice in Rome long before this time or Lauwers & Willekens (1994) whorefers to Colt (1844) whose thesis is that italians pick up their knowledgeof double entry bookkeeping at Alexandria, Constantinople or some othereastern cities50.
But many accounting historians, as Roover (1955), do not accept doubleregistration of a transaction, on time in credit and one time in debit, as asufficient condition to qualify the accounting system under study as adouble entry system. For instance, de Roover insists on the fact that alltransactions be twice recorded. ”This principle involves the existence of anintegrated system of accounts, both real and nominal, so that the books will balancein the end, record changes in the owner’s equity and permit the determination of
49For an analysis of the portrait painted by Jacopo de Barbari in 1495 which is exposedin many internet sites, it is worth to read McKinnon (1993).
50Recent investigations tend to confirm this hypothesis. For instance, Albraiki (1994)proves that at the beginning of the Mamluk period between 1250 and 1517, double entrybook keepin was already in use in Egypt and Syria. More than that, the old Cairo genizacollection — geniza meaning burial —, a collection of more than 200000 fragments foundin the Ben Ezra synagogue during its restoration in 1890, contains a fragment dated from1080 in the form of a journal and a four page account dated from 1134 listing both creditand debits — see Scorgie (1994).
37
profit and loss”.With this in mind, according to Roover (1955), the oldest discovered
record of a compete double-entry system is the one of the Messari — thetreasurers — accounts of the city of Genoa in 1340 because not only it con-tains debits and credits journalised in a bilateral form, but each transactionis recorded twice in the ledger.
Pacioli himself credited the first description of the system to BenedettoCotrugli, a Raguze merchant, who has written a book Delai Mercatura etdel Mercante Perfetto — Of Trading and the Perfect Trader — which has beenpublished a century later, but of which he was familiar.
The Summa de Arithmetica, Geometria, Proportioni et Proportionality —Everything about Arithmetic, Geometry and Proportion — has been written asa digest and guide to existing mathematical knowledge and bookkeepingwas only one of the five topics covered. The presence of bookkeeping inthis master piece and the fact that 37 short chapters entitled De Computisand Scripturis, were devoted to its study, acknowledged the fact that forPacioli it was a major and perfectly legitimate mathematical subject for histime, a true algebraic application.
But, not only for Pacioli was it a major subject. The proof come fromthe fact that it has not circulated as manuscript copies but as printed copies— as soon as it has been finished, it has been directly printed in 1494 fromthe Paganino de Paganini printing house. Yet, even if the fact to print drivethe reproduction price to a low level in comparison to the manual copy,in that times, printing could be achieved only to a high cost, and as theinvention was only in its thirteen decade, it must have been a recognizedurgency, to print such a book when so much manuscript where waitingto be printed. In all the cases, the book is an incunabulum that is to say aimmeasurable value book printed when printing was still in its cradle.
More than that, in choosing to print his master piece in Venice wherehe came to accomplish this project which is demonstrated by the fact thatsince 1486, he visited main courts and lectured mathematics at variousItalian universities such as Perugia, Florence, Rome and Napels where hetaught also military science when he occupied no recorded function inVenice, Pacioli was certainly aware of the fact that he could discuss withsome masters in accounting in the case where it happen to be necessary,and that it was the only place in the world where he was covered bysome author rights since they have been invented by the Serenissima in1474, even if it was only for the venetian states and for a 10 year duration.
38
Even if this copyright give no protection against the copy by hands, thereis no known hand copy of the Summa. It could seems strange becausethere was an abundance of scribes, whose work will last until the endof the 16th century because the strong resistance from some bibliophileswho preferred to possess an handwritten unique manuscript. According toSangster, Stoner & McCarthy (2007), the lack of pirated copies of the Summacould be explained in part by the presence of diagrams and marginal noteswhich could make the copy relatively unattractive.
And, as Padova was in the venetian states, and because in Padovathe University was independent of the Papacy, liberal fields were taughtopening a large market for the book which nevertheless has been writtenmainly for the merchants51,52.
The commercial success of the book was such that it was printed twotimes the same year and that his publisher Paganino de Paganini signedagain Pacioli for two others books which were published in 150953. In 1523,the son of Paganini published a new edition of the book.
In what concern the number of printed copies of the Summa a firstestimation of 300 printed copies by Antinori (1980) has been disallowed bySangster et al. (2007) on the basis that it does not take into account the sizeof the print-runs of the late 15th century. On this basis, in their opinion itwould be reasonable to infer that at least 500 copies were printed. But otherfactors, indicate that a greater number of copy could have been printed.
First of all some pages have been independently printed. It was thecase in 1502 and in 1509 certainly, in the case of the first date, avoid theexpiration of the 10-year copyright and in the case of the second one totake advantage of a 15 year copyright witch have been granted to Paciolihimself.
After some very convincing arguments, partially coming from the factthat the editor financed the publication of the 1523 second edition, Sangsteret al. (2007) arrive to the conclusion that more than 1000 and perhaps upto 2000 copies of the Summa where sell.
51Sangster et al. (2007) argue that because the book has no worked examples, he hasbeen written for merchants who need not them.
52
53De divina proportione which was controversial in the sense where the third part isa translation in italian of the treatise on the five regular solid written by Pierro dellaFrancesca and the translation of Euclid’s Elements.
39
2.2 The accountancy group
In accountancy an double-entry account can be defined as an ordered pairof number (d, c) ∈ Z2 where c is the credit and d is a debit.
(d, c) =Debit Credit
d c
In a newtonian accountancy, the gap between c and d is the balance or, ina more economic mood, the profit. In a newtonian world, we can add twoaccounts in such a way that if (d1, c1) is the first account and (d2, c2) is thesecond, we have : (d1, c2) + (d2, c2) = (d1 + d2, c1 + c2). We can also add threeaccounts in such a way that if the first is (d1, c1), the second (d2, c2), and thethird (d3, c3) we will have ((d1, c1) + (d2, c2)) + (d3, c3) = (d1, c1) + ((d2, c2) +(d3, c3)). We can also remark that as (0, 0) ∈ Z2, (d1, c1) + (0, 0) = (d1, c1). Inother terms, in accountancy, (0, 0) a neutral element.
Since if (d1, c1) ∈ Z2, −(d1, c1) ∈ Z2, in accountancy each accountancyhas obviously an inverse because (d1, c1) + [−(d1, c1)] = (0, 0). And last butnot least, the order in which one adds the accounts has no consequencesbecause (d1, c1) + (d2, c2) = (d2, c2) + (d1, c1).
In the mathematical terminology, an euclidian accountancy is associa-tive, it possess a neutral element and each account has an inverse and theaddition of two accounts is commutative. In short, an accountancy is whatmathematicians call an additive abelian group.
This as been noticed by Ellerman (1986) and incidentally by Lim (1966).Ellerman call it the Pacioli group54. As he remarks, a century earlier, the greatArthur Cayley — voir Cayley (1984[1896]) — has noticed by himself that ifmathematicians do not seriously looked to accountancy as a mathematicalobject it is because of its apparent simplicity55,56. Cayley was also the firstto linked double-entry book-keeping to the euclidian ratios :
54Ellerman use the notation [c||d] because it seems that it has been suggested by Paciolihimself.
55Cayley has been a lawyer for 14 years.56Augustus DeMorgan is the only other great mathematician who was interested in
accountancy — see DeMorgan (1869).
40
The Principles of Book-keeping by Double Entry constitutesa theory which is mathematically by no means uninteresting:it is in fact Euclid’s theory of ratios an absolutely perfectone, and it is only its extreme simplicity which prevents itfrom being a interesting as it would be otherwise — Cayley(1984[1896])
But each mathematical presentation of group theory display conjointlyan additive and a multiplicative group. So why is there no non-newtonianaccountancy based on the multiplicative one. Even there is only one likelyanswer to this question which is to say that our brain is delivered only tocompute additions and substractions, all other operations being acquiredwhich give a great advantage to addition over multiplication, one muststudy the possibility of a multiplicative accountancy on the basis thatthe conception of the modern accountancy system as been a very timeconsuming process and that nothing could have prevent this system tohave a distinct look from the one it finally takes.
If we call multiplicative accountancy an ?accountancy, it must operatewith ordered pairs (d, c) ∈ N? ×N?. In this non-newtonian system the ratiobetween debit and credit — i.e. : d/c — becomes the profit. We can multiplytwo accounts in such a way that if (d1, c1) is the first account and (d2, c2) isthe second, we will have (d1, c1)× (d2, c2) = (d1d2, c1c2). We can also multiplythree accounts (d1, c1), (d2, c2) and (d3, c3) and find that this is an associativeoperation — i.e. : ((d1, c1) × (d2, c2)) × (d3, c3) = (d1, c1) × ((d2, c2) × (d3, c3)).The ?accountancy is also commutative since (d1, c1) × (d2, c2) = (d2, c2) ×(d1, c1).The account (1, 1) ∈ N? × N? plays the role of the neutral elementfor the ?accountancy since (d, c) × (1, 1) = (d, c). Each account (d, c) has aninverse since (d, c) × (d−1, c−1) = (1, 1).
So as the newtonian accountancy is an abelian group, the non-newtonian?accountancy is also an abelian group — Ellerman (2010) call it a Paciolimultiplicative group by contrast with the newtonian accountancy which is aPacioli additive group.
3 Conclusion
Taking into account that it could be shown that ratios better compare twopositive quantities than differences, which has been discussed by manyRenaissance scholars including Galileo — Grossman & Katz (1972) — and
41
that from this it follows that growth phenomenon are better described bythe multiplicative point of view than by the additive one, one can considerthat the ?accountancy approach could at least been used to express balancesheets in the analysis of the economic growth. It’s what we have tried todo in Filip & Piatecki (2011).
References
Albraiki, S. (1994), Accounting in medieval Islam, Technical report, UnitedArab Emirates University.
Alexander, J. (2002), History of accounting, Technical report, Associationof Chartered Accountants in the United State.
Anderson, W. (1956), ‘Arithmetical computations in roman numerals’, Clas-sical Philology 51(3), 145–150.
Antinori, C. (1980), An Anomalous Edition of the ”Summa”, 1494 by LucaPacioli, Palatina Editrice.
Bell, H. (1937), The Price of Books in Medieval England, The Library.
Bergner, H. (1905), Handbuch der Kirchlichen Kunstaltertmer in Deutschland,Ch. Herm. Tauchnitz.
Black, R. (2004), ‘ecole et socit florence aux xivet xvsicle. Le tmoignage desricordanze’, Annales. Histoire, Sciences Sociales 59(4), 827–846.
Black, R. (2007), Education and Society in Florentine Tuscany. Teachers, Pupilsand Schools c. 1250-1500, Brill.
Bonaini, F. (1858), ‘Memoria unica sincrona di leonardo fibonacci, nova-mente scoperta’, Giornale Storico degli Archivi Toscani I, 5.
Brown, E. & Brunson, J. (2008), ‘Fibonaccis forgotten number’, The CollegeMathematical Journal 39(2), 112–120.
Cajori, F. (1909), A History of Mathematics, The MacMillan Company.
Cajori, F. (1950), A History of Elementary Mathematics.
42
Cardano, G. (1999[1666]), Omnia Opera, Johnson Reprint Corp.
Cayley, A. (1984[1896]), The Principles of Book-keeping by Double Entry, Cam-bridge University Press.
Clark, W. (1930), The Aryabhatiya of Aryabhata, The University of ChicagoPress.
Colt, J. (1844), The Art of Double-Entry Bookkeeping, Nafish & Cornish.
Davis, C. (1965), ‘Education in dante’s florence’, Speculum 40(3), 415–435.
DeMorgan, A. (1869), On the main principle principle of book-keeping, inJ. Watson, ed., ‘Elements of Arithmetics’.
Devlin, K. (2010), Recreational mathematics in Leonardo of Pisa’s LiberAbacci, Technical report, Standford University.
Dittmar, J. (2011), The welfare impact of a new good: The printed book,Technical report, Department of Economics, American University.
Drozdyuk, A. & Drozdyuk, D. (2010), Fibonacci, His Numbers and His Rab-bits, Choven Publishing Corp.
Durant, W. (1950), The Age of Faith, Simon and Schuster.
Ellerman, D. (1986), ‘Double entry multi-dimensionnal accouting’, Omega14(1), 13–22.
Ellerman, D. (2010), Generalized double-entry accounting: Showing whatis ”double” in the double-entry method.
Fibonacci, L. (2010), Tre Scritti Inediti Di Leonardo Pisano, Nabu Press.
Fibonacci, L. & Boncompagni, B. (1857-1862), Scritti di Leonardo Pisano:Mathematico del secolo decimoterzo, Tipografia delle Scienze Mathe-matiche e Fisiche, chapter Flos.
Fibonacci, L. & Sigler, L. (1987), Fibonaccis Liber Quadratorum, AcademicPress.
Fibonacci, L. & Sigler, L. (2003[1202]), Fibonacci’s Liber Abaci.
43
Filip, D. & Piatecki, C. (2011), A non-newtonian examination of the the-ory of exogenous economic growth, Technical report, Babes-BolyaiUniversity / Leo Orleans University.
Goetzmann, W. (2004), Fibonacci and the financial revolution, NBER Work-ing Paper 10352, Yale School of Management - International Centerfor Finance.
Goldthwaite, R. (1972), ‘Schools and teatchers of commercial arithmetics inRenaissance Florence’, Journal of European Economic History 1(2), 418–433.
Grimm, R. (2002), The autobiography of Leonardo pisano, Technical report,University of California at Davis.
Grossman, M. & Katz, R. (1972), Non-Newtonian Calculus, Lee Press.
Høyrup, J. (2010), Fibonacci – protagonist or witness? who taught catholicchristian europe about mediterranean commercial arithmetic?, Tech-nical report, Roskilde University, Section for Philosophy and ScienceStudies.
Ifrah, G. (1985[1981]), From One to Zero. A Universal History of Numbers.,Penguin Books.
Kaplan, R. (2000), The Nothing That Is. A Natural History of Zero, OxfordUniversity Press.
Kasir, D. (1931-1972), The Algebra of Omar Khayyam , New York, 1931.Reprinted AMS, 1972, Teachers College Press – AMS.
Kats, P. (1930), ‘A surmise regarding the origin of bookkeeping by doubleentry’, The Accounting Review 5(4), 311–316.
King, D. (2001), The Ciphers of the Monks, Franz Steiner Verlag.
Kuhn, T. (1962), The Structure of Scientific Revolutions, University of ChicagoPress.
Lakoff, G. & Nunez, R. (2000), Where Mathematics Comes From, Basic Books.
44
Lauwers, L. & Willekens, M. (1994), ‘Five hundred years of bookkeeping:A portrait of Luca Pacioli’, Tijdschrift voor Economic en ManagementXXXIX(3), 290–304.
Lay-Yong, L. (1966), ‘On the chinese origin of the galley method of arith-metical division’, The British Journal for the History of Science 3(1), 66–69.
Lehmann, P. (1936), ‘Bltter, seiten, spalten, zeilen’, Zentralblatt fr Bibliothek-swesen 53, 333–361 & 411–442.
Lim, R. (1966), ‘The mathematical propriety of accounting measurementsand calculations’, The Accounting Review 41(4), 642–641.
Littleton, A. (1933), Accounting Evolution to 1900, American Institue Pub-lishing co.
Machiavelli, B. (1954[v 1488]), Libro di ricordi, a cura di Cesare Olschi, Bib-lioteca Rara, Felice Le Monnier.
McKinnon, N. (1993), ‘The portrait of fra Luca Pacioli’, The MathematicalGazette 77(479), 130–219.
Murray, A. (1991), Reason and Society in the Middle-Age, Oxford UniversityPress.
Neugebauer, O. (1935), Mathematische Keilschrift-Texse, Springer-Verlag.
Piche, D. (1999), La Condamnation Parisienne de 1277, Librairie J. Vrin.
Pryor, J. (1977), ‘The origins of the commenda contract’, Specul 52(1), 5–37.
Ramanujacharia, R. & Kaye, G. (1913), ‘The Trisatika of Sridharacarya’,Bibliotheca Mathematica 3, 203–217.
Roover, R. (1955), ‘New perspectives ont the history of accounting’, TheAccounting Review 30(3), 405–420.
Rubinstein, M. (2006), A History of the Theory of Investments, John Wiley &Sons.
Sangster, A., Stoner, G. & McCarthy, P. (2007), The market for Luca Pacioli’sSumma Arithmetica, Technical report, 19th Accounting, Business &Financial History Conference, Cardiff.
45
Sarma, S. (2002), Rule of three and its variations in india, in Dold-Samplonius & alii, eds, ‘From China to Paris: 2000 Years Transmissionof Mathematical Ideas’, Franz Steiner Verlag.
Scorgie, M. (1994), Accounting fragments stored in the Old Cairo Genizah,Technical report, La Trobe University.
Seife, C. (2000), Zero. The Biography of a Dangerous Idea, Penguin Books ltd.
Smith, D. (1958), History of Mathematics, Dover Publications.
Smith, D. & Karpinski, L. (1911), The Hindu-Arabic Numeral, Ginn andCompany.
Stevin, S. (1935[1585]), ‘L’arithmetique’, Isis 23, 230–244.
Tangheroni, M. (2002), Un ponte sul Mediterraneo Leonardo Pisano, la scienzaaraba e la rinascita della matematica in Occidente, Polistampa.
Tangheroni, M. (2003), Pisa e il Mediterraneo. Uomini, merci, idee dagli Etruschiai Medici, Skira.
Turner, J. (1951), ‘Roman elementary mathematics. The operations’, TheClassical Journal 47(2), 63–108.
Ulivi, E. (2002), Scuole e maestri dabaco in italia tra medioevo e rinasci-mento, in E. Giusti & R. Petti, eds, ‘Un Ponte sul Mediterraneo.Leonardo Pisano, la scienza araba e la rinascita della matematica inOccidente’, Polistampa, pp. 121–159.
Velde, F. (2000), Notes on tuscan coinage, 12th c. 14th ce, Technical report,Federal Reserve Bank of Chicago.
Villani, G. (1906[1341]), Chronicle, Archibald Constable & Co Ltd.
Zervas, D. (1975), ‘The trattato dell’abbaco and andrea pisano’s design forthe florentine baptistery door’, Renaissance Quarterly 28(4), 483–503.
Zuccato, M. (2005), ‘Gerbert of Aurillac and a Tenth-century jewish channelfor the transmission of arabic science to the west’, Speculum 80(3), 742–763.
46
